In this paper, we consider estimating spot/instantaneous volatility matrices of high-frequency data collected for a large number of assets. We first combine classic nonparametric kernel-based smoothing with a generalised shrinkage technique in the matrix estimation for noise-free data under a uniform sparsity assumption, a natural extension of the approximate sparsity commonly used in the literature. The uniform consistency property is derived for the proposed spot volatility matrix estimator with convergence rates comparable to the optimal minimax one. For the high-frequency data contaminated by microstructure noise, we introduce a localised pre-averaging estimation method that reduces the effective magnitude of the noise. We then use the estimation tool developed in the noise-free scenario, and derive the uniform convergence rates for the developed spot volatility matrix estimator. We further combine the kernel smoothing with the shrinkage technique to estimate the time-varying volatility matrix of the high-dimensional noise vector. In addition, we consider large spot volatility matrix estimation in time-varying factor models with observable risk factors and derive the uniform convergence property. We provide numerical studies including simulation and empirical application to examine the performance of the proposed estimation methods in finite samples.

翻译：本文考虑对大量资产的高频数据估计现货/瞬时波动率矩阵。在均匀稀疏性假设下——这是文献中常用近似稀疏性的自然推广——我们首先将经典非参数核平滑与广义收缩技术结合，用于无噪声数据矩阵估计。所提出的现货波动率矩阵估计量具有一致收敛性，收敛速度与最优极小化速度相当。针对受微观结构噪声污染的高频数据，我们引入一种局部预平均估计方法，以降低噪声的有效强度。随后利用无噪声场景下开发的估计工具，推导出所构建现货波动率矩阵估计量的一致收敛速率。我们进一步将核平滑与收缩技术相结合，用于估计高维噪声向量的时变波动率矩阵。此外，针对具有可观测风险因子的时变因子模型，我们考虑大规模现货波动率矩阵估计并推导其一致收敛性质。通过包含模拟与实证应用的数值研究，我们检验了所提估计方法在有限样本下的表现。